(a) Give a direct proof of the fact that a^2-5a+6 is even for any integer a

Answer 1

Mathematical induction is a special way of proving things.

It has only two steps.

Step - 1) Statement is true for a = 1

Step - 2) one part being assume the statement is true for a =  k

prove the statement is true for " k + 1"

a² - 5a + 6

Check the equation for a = 1

1² - 5(1) + 6  = 1 - 5 + 6 = 2

2 is even integer.

For a = 1 it is true.

Assume the statement is true for a² - 5a + 6.

k² - 5k + 6 is even integer.

 (k + 1)² - 5(k + 1) + 6

= k² + 1 + 2k - 5k + - 5 + 6

= k² - 5k + 6 + 2k - 4

= (k² - 5k + 6) + 2(k - 2)

k² - 5k + 6 is even integer and 2(k - 2) is even.

Addition of even integer is even.

The statement is true for " k + 1".

Therefore, by the mathematical induction the statement is true.